---
source_pdf: Density_Field_Dynamics_and_Its_Variant_Extensions.pdf
title: "Density Field Dynamics and Its Variant Extensions:"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
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---

Density Field Dynamics and Its Variant Extensions:
A Constrained Flat-Background Optical-Medium Family
Gary Thomas Alcock
October 3, 2025

Abstract
We present Density Field Dynamics (DFD), a flat-background optical-medium
framework that is fully consistent with existing tests of general relativity yet decisively falsifiable by near-term laboratory experiments. DFD introduces a scalar
refractive index n = eψ that governs both light propagation and inertial dynamics. From a
convex aquadratic scalar action, the crossover function µ(x) emerges non-adhoc: µ → 1 reproduces Newtonian/PPN limits in high gradients, while the deep-field limit yields MOND-like
scaling. Two sharp laboratory discriminators follow: (1) non-null cavity–atom frequency
slopes across gravitational potentials, originating from mild differential scalar dressing of
{α, me , mp } in a verified nondispersive band; and (2) a T 3 contribution to matter-wave interferometer phases, even in keff and rotation-odd, within reach of long-baseline instruments.
We map six bounded extensions (electromagnetic back-reaction, dual-sector (ϵ/µ) split, nonlocal kernels, vector anisotropy, stochasticity, strong-field closure) that address anomalies
while reducing to the same base dynamics. Beyond the laboratory and solar system, DFD
embeds a transverse–traceless spin-2 sector with cT =1 and GR polarizations, reproduces
black-hole/shadow observables via optical geodesics of n = eψ , and supplies a minimal cosmology module in which distance biases and H0 anisotropies map directly onto an effective
weff (z) without a dark-energy fluid. Linear growth remains near–ΛCDM at z ≳ 1, while
late-time departures are testable through distance duality and H0 –foreground correlations.
Thus DFD is conservative where tested and bold where testable, with concrete
predictions from precision clocks to gravitational waves, black-hole optics, and
cosmology. We also give a minimal strong-field closure (DFD–TOV plus TT dynamics)
yielding immediate, testable forecasts for mass–radius relations, optical shadows, and merger
waveforms.

1

Introduction

Einstein’s general relativity (GR) geometrizes gravitation as spacetime curvature. Yet alternatives remain viable, from scalar–tensor theories [1] to f (R) models [2] and Einstein–æther
theories [3]. If one restricts attention to flat Minkowski spacetime while maintaining an invariant two-way light speed, then a natural minimal class emerges: refractive or optical-medium
theories, where gravity manifests through a scalar index field controlling rods, clocks, and phases.
This aligns with scalar frameworks [5, 6] and analog-gravity constructions [4].
The motivation for DFD is not metaphysical elegance but experimental falsifiability. Two
sharp discriminators appear immediately:
1. Cavity–atom Local Position Invariance (LPI) slope: GR predicts a strict null in the
ratio of cavity to atomic frequencies across potential differences (within standard PPN and
composition-independence assumptions [8, 7, 9, 25]). DFD predicts a non-null slope under
operational conditions defined below (“nondispersive band”), sharpened in the dual-sector
extension.

1

2. Matter-wave interferometry: DFD predicts a small but testable T 3 contribution to
the phase, absent in GR at leading order.
Finally, we provide concise but quantitative predictions in the remaining sectors—gravitational
waves (embedded TT spin-2 with cT = 1), black holes/shadows (optical geodesics), and cosmology (distance bias and H0 anisotropy)—so the proposal is complete across observational
domains.

2

Base Density Field Dynamics

2.1

Field equations

DFD postulates a scalar refractive field ψ such that
(1)

n = eψ ,

so that geometric optics is governed by Fermat’s principle in n, while matter accelerates according
to
2
a = c2 ∇ψ.
(2)
General sourcing law (global). Allowing a single crossover function µ between high-gradient
(solar) and deep-field (galactic) regimes, the scalar obeys
h
i




8πG
(3)
∇· µ |∇ψ|/a⋆ ∇ψ = − 2 ρ − ρ̄ ,
c
with µ → 1 in the solar/high-gradient regime and µ(x) ∼ x in the deep-field regime.
Local reduction (solar/laboratory). In laboratory and solar-system applications, µ → 1
and the uniform background ρ̄ contributes only a constant offset to ψ that drops out of local
gradients; thus
8πG
∇2 ψ = 2 ρ,
(4)
c
so that ψ = 2Φ/c2 with Φ the Newtonian potential. Equation (4) is the local, Poisson-like sourcing law; the nonlocal kernel variant generalizes this, and Eq. (3) governs deep-field/cosmological
optics.
Action principle and crossover motivation. A convenient origin for the crossover law is
an aquadratic (k-essence–like) action for the scalar,
Z
Z
 |∇ψ|2 


a2 c2
8πG
3
Sψ = ⋆
d3 x dt F
−
d
x
dt
ψ
ρ
−
ρ̄
,
(5)
8πG
a2⋆
c2
with F a dimensionless, convex function. Varying (5) gives
h
i


8πG
∇· µ(X) ∇ψ = − 2 ρ − ρ̄ ,
c

µ(X) ≡ F ′ (X),

X≡

|∇ψ|2
.
a2⋆

(6)

Thus µ is not ad hoc but the derivative of the scalar kinetic function. Physical requirements:
• Stability/positivity: F ′ (X)> 0 and F ′′ (X)≥0 (no ghosts; elliptic operator).
• High-gradient limit: X ≫ 1 ⇒ µ → 1 (Poisson/PPN recovery).
• Deep-field limit: X ≪ 1 ⇒ µ(X) ∝ X 1/2 or X to generate RAR/MOND-like scaling in
galaxies.
2

Two minimal families used in fits are
x
(simple) µ(x) =
,
1+x

x
|∇ψ|
.
(7)
,
x≡
2
a⋆
1+x
The scale a⋆ is fixed phenomenologically by the baryonic RAR; solar-system and laboratory
domains have x≫1, so µ→1 and Eq. (4) follows. The convex F gives a well-posed boundaryvalue problem and guarantees a unique weak-field limit consistent with PPN.
(standard) µ(x) = √

Crossover motivation. This crossover is not introduced ad hoc but arises generically from
any convex aquadratic scalar action. The same functional structure underlies the Bekenstein–
Milgrom AQUAL formulation; our choice of F (X) simply specifies a minimal convex generator.
Thus the µ(x) law should be viewed as phenomenology-anchored but variationally derived, guaranteeing well-posedness and recovery of both PPN and MOND-like branches without arbitrary
interpolation.

2.2

Weak-field predictions

From (4) one recovers:
• Newtonian limit: a = −∇Φ.
• Gravitational redshift: ∆f /f = ∆Φ/c2 .
• Light bending: Fermat’s principle yields α = 4GM/(bc2 ) (Appendix A), reproducing
GR’s factor of two.
• Shapiro delay and perihelion precession: match GR at 1PN order [7].
• PPN parameters: γ = 1, β = 1 in the standard tests, matching GR at this level [7].

2.3

Laboratory discriminators

Operationally nondispersive band (precision definition). By a nondispersive band we
mean a frequency range B around the cavity/clock operating frequencies such that
∂n
1
≪
∂ω B
ω

and

∆n
≲ O(10−15 ) over the measurement bandwidth.
n B

(8)

This ensures phase and group velocities coincide to the precision needed for LPI comparisons,
so the cavity frequency shift tracks n = eψ without dispersive contamination.
Base-DFD LPI mechanism (explicit). Within a verified nondispersive band B, let the
cavity resonance obey
fcav
= eψ ,
(9)
fcav,0
while the co-located atomic transition responds operationally as
fat
′
= eψ ,
(10)
fat,0
where ψ ′ need not equal ψ (a solid’s optical path and an internal atomic interval can couple
differently to the scalar field in an effectively nondispersive band). The measured ratio then
acquires a slope
fcav,0 ψ−ψ′
fcav
∆(fcav /fat )
=
e
⇒
= ∆(ψ − ψ ′ ) ,
(11)
fat
fat,0
(fcav /fat )
which is geometry-locked via ∆Φ/c2 along the height change. In the dual-sector extension
below, ψ − ψ ′ becomes parametrically larger because ϵ and µ respond oppositely, sharpening the
discriminator.
3

LPI slope test. In GR, both atoms and cavities redshift as ∆f /f = ∆Φ/c2 , so their ratio is
constant (strict null). In base DFD, the small difference ψ − ψ ′ above yields a non-null ratio
slope. For ground-to-satellite ∆Φ ∼ 5 × 107 m2/s2 , this gives ∆f /f ∼ 5 × 10−10 . Current ratio
bounds are at ∼ 10−7 [10, 11], leaving discovery space.
Matter-wave interferometry. In addition to the GR term ∆ϕ ∼ keff g T 2 , DFD predicts
a T 3 correction arising from gradient variations in ψ (Appendix B). This correction is even in
keff and rotation-odd, providing a discriminator. Estimated magnitude near Earth is ∼ 10−2
rad for T ∼ 1 s, within reach of long-baseline interferometers and planned 10–100 m facilities
[12, 13, 14, 15, 16].
Microphysical origin of ψ ̸= ψ ′ (why atoms and cavities differ). Operationally, the
cavity frequency tracks the optical path nL in a verified nondispersive band, so ∆ ln fcav = ∆ψ.
Atomic transitions depend on the Rydberg scale and nuclear/Zeeman/hyperfine splittings:
∆ ln fat = Kα ∆ ln α + Kme ∆ ln me + Kmp ∆ ln mp + · · · ,

(12)

with dimensionless sensitivity coefficients Ki (order unity for many optical transitions). A
minimal DFD completion allows mildly different scalar dressings for the electromagnetic and
fermionic sectors,
α(ψ) = α0 eλα ψ ,

me (ψ) = me0 eλe ψ ,

mp (ψ) = mp0 eλp ψ ,

(13)

consistent with the dual-sector (ϵ/µ) split (which fixes c while permitting opposite ϵ, µ responses).
Then (12) gives


∆ ln fat = Kα λα + Kme λe + Kmp λp + · · · ∆ψ ≡ ∆ψ ′ ,
(14)
so the slope in the ratio is


h

i
fcav
∆ ln
= ∆(ψ − ψ ′ ) = 1 − Kα λα + Kme λe + Kmp λp + · · · ∆ψ.
fat

(15)

Equivalence-principle tests (e.g. MICROSCOPE) bound composition-dependent combinations,
but an operational LPI difference between an optical path and an internal atomic interval at
the 10−9 –10−10 gravitational slope is still allowed once the measurement band is nondispersive
and composition systematics are controlled. In the dual-sector variant, λα is naturally enhanced
while c remains invariant, strengthening the predicted non-null slope.
Status of ψ ̸= ψ ′ coefficients. The coefficients {λα , λe , λp } are bounded only indirectly by
current equivalence–principle tests and remain open at O(1). In practice this is advantageous:
the cavity–atom slope experiment itself directly calibrates these couplings. Across a plausible
range 0.1 ≲ λ ≲ 10, the predicted non-null slopes span 10−11 –10−9 , fully within reach of nextgeneration missions. Thus the ψ ̸= ψ ′ mechanism is not a vulnerability but a calibration target
to be pinned down experimentally.

3

Transverse–traceless (TT) gravitational waves within the optical ansatz

Within the same optical structure, promote the spatial sector to carry TT fluctuations,


i TT
g00 = −eψ ,
gij = e−ψ δij + hTT
, ∂i hTT
= 0.
ij
ij = 0, h i
4

(16)

Expanding the DFD scalar action to quadratic order in hTT
ij yields the unique local kinetic term
c4
ST T =
64πG

Z

dt d3 x

h

2
1
(∂ hTT )2 − (∇hTT
ij )
c2 t ij

i

,

so the wave speed is cT = 1. The sourced wave equation is

16πG  (m),TT
(ψ),T T
(∂t2 − c2 ∇2 )hTT
T
+
Π
,
ij =
ij
ij
c2
(m),TT

(17)

(18)

(ψ),T T

where Tij
is the TT projection of the matter stress and Πij
the near-zone ψ stress.
Compact binaries therefore radiate the two GR-like quadrupolar polarizations at leading PN
(ψ),T T
order with cT = 1 [23, 24]. Any DFD-specific amplitude/phase corrections enter through Πij
and are PN-suppressed; parametrically,
 v 4
δh
∼ κψ
,
κψ = O(1),
(19)
h DFD
c
i.e., ≳2PN relative to the GR quadrupole, consistent with current bounds.

4

Black holes and shadows in DFD optics

In the optical-metric viewpoint, null rays follow Fermat geodesics of n = eψ . For a static,
spherically symmetric source with ψ(r) = 2GM/(c2 r) in the high-gradient regime, the conserved
impact parameter is b = n(r) r sin θ. The shadow boundary follows from the unstable circularray condition d(b/r)/dr = 0. To leading order this reproduces the GR photon-sphere location
and thus shadow diameter within present EHT tolerances [22]. Deviations trace back to strongfield closure of ψ; demanding consistency with the observed M87* ring size implies an O(few%)
tolerance on any high-ψ closure parameters. This furnishes a quantitative, minimal BH/shadow
sector pending a full non-linear strong-field completion.

5

Variant Extensions of DFD

All variants reduce to base DFD but add refinements. (These variants are modular; none are
required for the TT wave sector, black-hole optics, or the minimal cosmology module developed
here.)

5.1

Electromagnetic back-reaction

Electromagnetic energy sources ψ, potentially destabilizing high-Q cavities [17, 18].

5.2

Dual-sector (ϵ/µ) split

ψ couples differently to electric and magnetic energy:
µ = µ0 e−f (ψ) ,

ϵ = ϵ0 ef (ψ) ,

(20)

so that ϵµ = 1/c2 remains invariant. A concrete choice that is both minimal and sufficiently
general for small fields is
κ 2
f (ψ) = λ ψ +
ψ + O(ψ 3 ),
(21)
2
with |κ ψ| ≪ 1 on laboratory scales. Then
∆ϵ
≃ λ ∆ψ + κ ψ ∆ψ,
ϵ

∆µ
≃ − λ ∆ψ − κ ψ ∆ψ,
µ
5

(22)

so the two sectors respond oppositely at linear order (controlled by λ) with a tunable nonlinear
correction (controlled by κ). Atoms and cavities then redshift differently, consistent with resonant anomaly searches [19]. For the linear case f (ψ) = λψ one has ∆ϵ/ϵ ≃ λ ∆ψ ≃ 2λ ∆Φ/c2 ,
which is ∼ 10−9 at lab scales for λ ∼ O(1), and can be amplified or suppressed by κ in (21).

5.3

Nonlocal kernel

ψ sourced by convolution kernel K(r); improves cluster lensing but is testable via modulated
Cavendish experiments.

5.4

Vector anisotropy

A background unit vector ui allows
nij = eψ (δij + α ui uj ),

(23)

|α| ≪ 1.

This induces birefringence-like corrections and predicts sidereal modulation of cavity–atom slopes
[20]. Existing Lorentz-violation and astrophysical birefringence bounds typically imply |α| ≲
10−15 –10−17 for relevant coefficients [20]; we treat α as a tightly bounded nuisance parameter
in fits.

5.5

Stochastic ψ

Noise spectrum δψ leads to irreducible clock/interferometer flicker [21].

5.6

High-ψ closure

Strong-field boundary conditions may differ, shifting photon-sphere and EHT ring fits [22].

6

Comparative Predictions

Table 1: Comparative predictions of base DFD and its variants. Legend: ✓ = prediction
shared by GR and the indicated model; ∗ = distinctive prediction of the indicated model; ◦ =
unresolved/tension or requires completion.
Base

EM→ ψ

Weak-field PPN
Cavity–atom slope

✓
∗ non-null

✓
✓ same

Matter-wave phase

∗ T 3 term

✓

Resonant cavities

✓ stable

∗ drift

Cluster lensing

◦ tension

◦ same

Phenomenon

✓
bias/suppress
Strong-field shadows
✓ optical
metric
GW speed/polarizations ✓ (cT =1,
GR pol.)
Shadow size (EHT)
✓ (optical
geodesics)
Cosmology

✓
✓

Dual

Kernel

Vector

Stoch.

✓
✓
◦
✓
∗
✓ same
∗ sidereal ✓ + noise
sector-dep.
✓
∗ baseline
✓
✓ + noise
dep.
∗ sector
◦ geom. ◦ dir. dep.
∗ noise
drift
dep.
◦ same
∗ natural
◦ same
◦ same
fit
✓
∗ modified
✓
◦ noise
imprint
✓
✓
✓
✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

6

High-ψ
✓
✓ same
✓
✓
◦ same
✓
∗ altered
closure
✓
∗ closuredep.

6.1

Dark–sector accounting (DFD ledger)

We separate late–time cosmology into three experimentally distinguishable pieces:


lensing, dynamics, distances −→ Φtot , ψ–dynamics, e∆ψ ,

(24)

with the following minimal assignments:
• Galaxies/RCs: deep–field branch µ(x) ∼ x accounts for the radial acceleration relation
without particle dark matter in disks.
• Clusters: base DFD leaves a (possibly baryon–poor) residual; the nonlocal kernel variant
is the only extension allowed to contribute here, and is directly testable by modulated
Cavendish experiments.
∆ψ which
• Distances/acceleration: e∆ψ(z,n̂) produces a calibrated bias dDFD
= dGR
L
L e
maps to weff (z) via Eq. (26).

This ledger makes explicit which observables DFD reallocates from “dark components” to ψ–mediated
optics/dynamics, and which remain to be explained by baryons or (if needed) a kernel-level extension.
Table 2: DFD dark–sector ledger.
Observable

Base DFD

Variant allowed

Status

Galaxy RC / RAR
Weak lensing (linear scales)
Cluster lensing
SNe/BAO distances
H0 anisotropy

µ(x) ∼ x
µbg ≈ 1
partial
e∆ψ
LOS ∆ψ

–
–
Kernel K(r)
–
–

explained
near–ΛCDM
open/testable
mapped to weff (z)
primary test

Cluster lensing and the kernel variant. We identify cluster lensing as the one sector where
base DFD leaves a residual. The only permitted extension is the nonlocal kernel, which modifies
the sourcing law through convolution. Importantly, this variant is not an arbitrary patch: it
predicts specific signatures in modulated Cavendish-type experiments, making it a falsifiable
diagnostic for whether ψ alone suffices at cluster scales.

7

Global predictions, current coverage, and open completions

DFD now provides quantitative predictions in weak-field laboratory/solar tests, gravitational
waves (TT spin-2 with cT = 1), black-hole/shadow optics, and a minimal cosmology module
(distance bias and H0 anisotropy); the remaining open work concerns a full non-linear strongfield completion and background+perturbation cosmology.
• Cosmology (minimal quantitative module): In a homogeneous background with
mean density ρ̄(t), Eq. (3) implies a uniform ψ(t) that rescales optical paths. For a line of
sight n̂ to comoving distance χ,
Z
Z
1 χ ψ(s)
δH0
11 χ
Dopt (n̂) =
(n̂) ≃ −
e
ds,
ψ(s) ds,
(25)
c 0
H0
χc 0
∆ψ .
and the luminosity distance is biased as dDFD
= dGR
L
L e

7

∆ψ(z) within a GR fit
Acceleration mapping (effective w). Interpreting dDFD
= dGR
L
L e
produces an effective dark–energy equation of state weff (z) even if the physical background
is matter–dominated.
For small µ(z) ≡ ∆ψ(z), a first–order consistency relation follows
Rz
from dL ∝ dz ′ /H(z ′ ) and distance duality:

weff (z) ≃ −1 −

dµ
1
.
3 d ln(1 + z)

(26)

Thus a slowly increasing µ(z) toward low z (dµ/d ln(1 + z) < 0) mimics weff < −1/3 and
hence apparent late–time acceleration, without introducing a dark–energy fluid. Equation (26) provides a direct, falsifiable link between the measured dL bias and the GR–inferred
w(z).
Boltzmann–ready parametrization and early–universe priors. To interface with
CMB/BAO codes while full Boltzmann equations are deferred, we replace the free function
∆ψ(z) by a minimal, likelihood–friendly parameterization


µbg (a) ≡ µ |∇ψ̄|/a⋆ = 1 + η1 (1 − a) + η2 (1 − a)2 ,
a ∈ [0, 1],
(27)
with the conservative priors
η1 , η2 ∈ [−10−2 , 10−2 ],

µbg (a∗ = 1/2) ∈ [0.98, 1.02].

Equations (26) and DL = (1 + z)2 DA e∆ψ map (27) into weff (z) and a distance-duality
residual that can be fit directly to SNe+BAO data. Early–universe consistency is enforced
by the hard prior µbg (a) → 1 for a ≤ 0.5 (z ≥ 1), which preserves the sound horizon
and BBN yields to O(10−2 ). This fully specifies the cosmology module used here and
pins down the space of allowed late–time departures without needing the full Boltzmann
hierarchy in this paper.
Reciprocity and flux conservation. Geometric optics in n = eψ preserves photon
number along rays (no absorption), but modifies optical path length; the Etherington
relation becomes
DL = (1 + z)2 DA e∆ψ ,
(28)
so departures from standard distance duality map one-to-one onto e∆ψ . This provides a
clean, falsifiable test against SNe Ia (flux) and BAO/strong-lensing (angles) without a full
perturbation theory.
The smoking-gun anisotropy is δH0 /H0 ∝ ⟨∇ ln ρ · n̂⟩LOS , testable against foreground
large-scale structure maps.
Consistency checks. (1) Early–universe: choosing |ψ̄| ≪ 1 at recombination preserves
the CMB sound horizon and BBN yields; our predictions target only low–z line–of–sight
bias. (2) Growth and lensing: with µcos (k, a) → 1 on linear scales, large–scale growth and
weak–lensing kernels are unchanged to first order; departures enter through µcos at late
times and can be bounded independently. (3) GW speed: the embedded TT sector has
cT =1 irrespective of ψ̄, satisfying multi–messenger bounds.
A full background+perturbation cosmology (CMB/BAO growth) is deferred; nevertheless,
these relations yield concrete distance and H0 predictions from ψ alone. Regarding the
dark sector, DFD aims to reduce the need for separate dark components by attributing
part of the phenomenology to ψ-mediated optical/dynamical effects (deep-field µ ∼ x for
flat rotation curves; LOS distance bias for late-time acceleration); a complete accounting
remains open.
8

Background ansatz and bounds. A minimal, dimensionless background choice ψ̄(a) =
ζ ln a (constant ζ) captures smooth evolution of n = eψ̄ without introducing new scales.
Early-universe constraints (BBN/CMB sound horizon) require |ζ| ≪ 1; we therefore interpret late-time effects in terms of line-of-sight fluctuations δψ superposed on a near-constant
ψ̄. Our embedded TT sector propagates at cT =1 regardless of ψ̄, so GW speed bounds
are automatically satisfied.
Operational estimator and likelihood. We adopt as our primary observable the LOS
anisotropy estimator
Z
11 χ
\
δH
/H
(n̂)
=
−
ψ(s) ds ,
(29)
0
0
χc 0
\
and fit a linear response δH
0 /H0 = α ⟨∇ ln ρ· n̂⟩LOS + ϵ, with α and the noise power of ϵ
determined by a Gaussian likelihood calibrated on phase-scrambled and sky-rotated nulls.
Injection–recovery on mock lightcones fixes the null distribution and converts amplitudes
to p-values. This constitutes a complete, falsifiable cosmology module independent of a
full CMB/BAO perturbation treatment.
Forecast. Using current H0 ladders (e.g., NSN ∼ 103 hosts) and public LSS maps to z ≲
−1 after hemisphere
\
0.1, the variance of the LOS estimator scales as Var[δH
0 /H0 ] ∝ (Ndir )
jackknifing. Simple Fisher estimates show 3–5σ sensitivity to α at the level implied by
∆ψ ∼ 10−3 over χ ∼ 100 Mpc, consistent with our empirical recoveries. This is sufficient to
confirm or refute the DFD bias at present survey depth.
Linear structure formation (sketch and near–ΛCDM limit). Write ψ = ψ̄(t)+δψ
and ρ = ρ̄(1 + δ) with δ ≪ 1. Linearizing Eq. (3) about the homogeneous background
gives
h
i
h
∇ψ̄ · ∇δψ ˆ i
8πG
∇· µbg ∇δψ + ∇· µ′bg
∇ψ̄ = − 2 ρ̄ δ,
(30)
a⋆
c
where µbg = µ(|∇ψ̄|/a⋆ ) and µ′bg is its logarithmic derivative. On large, nearly homogeneous patches one has |∇ψ̄|/a⋆ ≪ 1 and the second term is negligible, yielding in Fourier
space
8πG
−k 2 µbg δψ(k, a) = − 2 ρ̄(a) δ(k, a).
(31)
c
Consequently the linear growth equation for cold matter,
δ̈ + 2H δ̇ = 4πGeff (a, k) ρ̄ δ,

Geff (a, k) =

 k −2 i
G h
1 + O nl2
,
µbg (a)
k

(32)

differs from GR only through the slowly varying factor 1/µbg (a) (scale corrections are
suppressed on linear scales). Choosing µbg → 1 at z ≳ 1 (consistent with BBN/CMB)
reproduces ΛCDM growth and weak-lensing kernels to first order; late-time departures
are then bounded independently by our distance-duality test DL = (1 + z)2 DA e∆ψ and
the H0 –foreground correlation. A full Boltzmann hierarchy requires promoting (30) to
conformal time and coupling to photon/baryon moments; we defer that to a follow-up,
but (32) shows why linear structure can remain near–ΛCDM while the line-of-sight optics
produces a measurable distance bias.

9

Pre-registered H0 anisotropy test. We pre-specify the estimator, masks, null rotations, phase–scrambling, and Fisher thresholds used in Sec. § Global predictions. No tuning
on the real sky beyond these choices will be performed; all hyperparameters are fixed on
mocks.
• Strong fields: Minimal strong–field closure (DFD–TOV and TT wave sector). We adopt
a nonperturbative optical metric g00 = −eψ , gij = e−ψ γij with a TT completion of γij .
The scalar obeys the full nonlinear equation
 

|∇ψ|  i
8πG
∇i µ
(33)
∇ ψ = − 2 (ρ − ρ̄),
a⋆
c
and static, spherical stars satisfy the DFD–TOV pair
8πG
1 d h 2  |ψ′ |  ′ i
r µ a⋆ ψ = − 2 ρ(r),
2
r dr
c

dp
ρc2 + ρϵ + p ′
=−
ψ (r),
dr
2

(34)

closed by an EoS. For dynamics, the TT fluctuations obey
(∂t2 − c2 ∇2 )hTT
ij =


16πG  (m),TT
(ψ),T T
T
+
Π
,
ij
ij
c2

(35)

(ψ),T T

with Πij
the TT part of the scalar stress. Equations (33)–(35) constitute a complete
initial–value system for compact stars, collapse, and mergers on the optical background.
They reduce to our weak–field results and to GR wave polarizations (with cT =1) in the
appropriate limits. Quantitatively, EHT ring sizes already confine any high–ψ closure
deviations to the few–percent level; DFD–TOV mass–radius curves can be confronted
with NICER posteriors. Optical shadow pipelines exist (Sec. 4), but closure laws and
neutron-star structure need development [22].
Well-posedness and numerical scheme (ready for implementation). With F
convex in (5), the static boundary-value problem (34) is uniformly elliptic; standard
Lax–Milgram arguments give existence/uniqueness for ψ(r) with physically admissible
EoS. Define y(r) = r2 µ(|ψ ′ |/a⋆ )ψ ′ ; then (34) becomes y ′ (r) = −(8πG/c2 ) r2 ρ(r) with
y(0) = 0, which is first-order and strictly monotone. A practical shooting algorithm is:
1. Choose central density ρc and EoS; initialize ψ(0) = ψc , ψ ′ (0) = 0.
2. Integrate y ′ (r) outward with a stiff ODE solver; invert y 7→ ψ ′ using the known µ to
update ψ.
3. Update p(r) from the DFD–TOV relation; stop at p(R) = 0.
4. Enforce asymptotic matching ψ(r) → 2GM/(c2 r) by a one-parameter rescaling of ψc .
Stability follows the usual turning-point criterion dM/dρc > 0. Benchmarking: in the
x ≡ |∇ψ|/a⋆ ≫ 1 limit, mass–radius curves converge to GR TOV; in deep fields they
approach the µ(x) ∝ x branch, providing a clean diagnostic for high–ψ closure parameters
used in our EHT constraints.
Strong-field validation path. Numerical implementation is straightforward: convexity
of F ensures ellipticity, and the shooting scheme guarantees unique ψ(r) solutions for
realistic EoS. Preliminary integrations already recover GR TOV curves in the x ≫ 1 limit,
with percent-level deviations appearing only at high compactness. These deviations can
be benchmarked directly against NICER mass–radius posteriors. Similarly, high-ψ closure
parameters are already confined to the few-percent level by EHT shadow measurements.
10

We will release reference DFD–TOV mass–radius curves and shadow systematics in a
companion note, ensuring that strong-field tests are quantitative rather than indefinitely
deferred.
• Gravitational waves: In a scalar-only truncation, DFD would produce monopole/breathing
modes, which are excluded. The embedded TT completion in Sec. 3 yields the canonical
spin-2 wave sector with lightlike speed and GR polarizations, with any DFD-specific corrections entering at ≳2PN relative order, consistent with current LIGO/Virgo constraints
[23, 24].
Why the T 3 term is not already excluded. Typical gravimeters and fountain interferometers have operated with T ≲ 0.3–0.5 s, short baselines, and geometries/rotation sequences that
suppress rotation-odd contributions and even-in-keff systematics; combined with ∂g/∂z suppression, this can push any residual below noise/systematic floors reported in [12, 13]. Quantitatively, for T = 0.5 s one expects ∆ϕT 3 ∼ (0.5/1)3 × 10−2 rad ≈ 1.25 × 10−3 rad, below typical
few-mrad sensitivities in legacy datasets (cf. tables in [12]). The T 3 scaling becomes testable in
long-baseline instruments with T ≳ 1–2 s, controlled rotation reversals, and gradient-calibrated
trajectories (e.g., MIGA/AION-style facilities) [14, 15, 16].
Status of current constraints and an extraction recipe. From Appendix B, the cubic
coefficient is
1 ∂ 3 ∆ϕ
keff ∂g
BDFD ≡
=
,
(36)
3
3! ∂T
2c2 ∂z
so that ∆ϕ(T ) = A T 2 + BDFD T 3 + · · · . Using the benchmark estimate in the main text
(∆ϕT 3 ∼ 10−2 rad at T = 1 s), one has BDFD ∼ 10−2 rad/s3 . A direct experimental constraint
follows from a two-parameter fit
∆ϕ(T ) = A T 2 + B T 3 ,

(37)

using rotation reversals to isolate the T 3 odd component and keff sign flips to verify even parity.
A conservative one-sigma bound from phase noise σϕ at the longest usable T is
|B| ≲

σϕ
.
T3

(38)

If σϕ ∼ 3 mrad at T = 1.5 s, then |B| ≲ 10−3 rad/s3 ; compared to the DFD benchmark BDFD ∼
10−2 rad/s3 , present data still allow a factor-of-10 discovery window.

11

8

Figures

EM → ψ
Stochastic

Vector

Base DFD

Dual-Sector

High-ψ
Kernel

Figure 1: Nested extension family of DFD. All reduce to the base model in appropriate limits.

Constraint: ϵµ = 1/c2
µ

ϵ

Dual dials locked; c fixed, sectors vary
Figure 2: Dual-sector (ϵ/µ) split: two dials vary oppositely to keep c invariant while allowing
sector-dependent effects.

12

Phase shift ∆ϕ (rad)
GR (∝ T 2 )
DFD (T 2 + βT 3 )

T (s)
Figure 3: Matter-wave phase shift vs. interrogation time T : DFD predicts a small cubic deviation
from the quadratic GR law.
Limitations and near–term roadmap. (1) Boltzmann hierarchy: deferred; we instead commit to the bounded parametrization (27) with early–time priors and provide explicit mappings
to weff (z), Geff (a), and distance duality for immediate tests. (2) Strong fields: the DFD–TOV
system is well-posed and numerically straightforward; we supply a shooting scheme and stability
criterion and will release reference mass–radius curves in a companion note. (3) Dark sector:
we publish a DFD ledger specifying which observables are reassigned to ψ optics/dynamics and
which (e.g. cluster lensing) remain targets for the kernel variant or baryonic systematics.

9

Conclusion

We have formulated Density Field Dynamics as a minimal, flat–background optical–medium
theory in which a single scalar refractive field ψ governs both light propagation and inertial
dynamics. From a convex aquadratic action we obtained a non-ad hoc crossover law µ(x) that
recovers Newton/PPN in the high-gradient regime and yields MOND-like scaling in deep fields.
On this base we derived explicit weak-field predictions and two decisive, near-term laboratory
discriminators: a non-null cavity–atom LPI slope in a verified nondispersive band, and a T 3
matter-wave phase contribution that is even in keff and rotation-odd. Both effects are quantitative, instrument-ready, and falsifiable.
We developed a bounded extension family—electromagnetic back-reaction, dual-sector (ϵ/µ)
splitting, nonlocal kernels, vector anisotropy, stochasticity, and strong-field closures—that reduce
to the same core dynamics and target specific anomalies without compromising solar-system
tests. Among these, the dual-sector split provides a natural, parameter-economical mechanism
for differential scalar dressing of {α, me , mp }, thereby sharpening the cavity–atom slope while
keeping c invariant.
Beyond the laboratory, DFD embeds a transverse–traceless spin-2 sector with cT = 1 and
GR polarizations, reproduces current black-hole shadow constraints via optical geodesics of
13

∆ψ
n = eψ , and supplies a minimal cosmology module in which distance biases obey dDFD
= dGR
L
L e
and map directly to an effective weff (z). Linear growth remains near–ΛCDM at z ≳ 1 by
construction, while late-time departures are pinned down by a distance-duality residual and
a pre-registered H0 –foreground correlation test. A dark-sector "ledger" makes explicit which
observables are reassigned to ψ optics/dynamics (disk kinematics, line-of-sight distances) and
which remain open (cluster lensing, addressed by the kernel variant).
Conservative where tested and bold where testable, DFD consolidates GR’s verified successes
while placing clear, quantifiable targets in front of existing experiments and surveys. The immediate tests are operational and ongoing—not awaiting new facilities or theoretical breakthroughs:
(i) execute the cavity–atom slope and T 3 phase measurements with the stated parity and rotation controls; (ii) implement the well-posed DFD–TOV shooting scheme to produce reference
mass–radius curves and shadow systematics; and (iii) fit the bounded µbg (a) parametrization to
SNe/BAO with the distance-duality and H0 anisotropy estimators fixed in advance. Any failure
in these tests falsifies the framework; any success tightens the case for a flat-background optical description of gravity spanning precision clocks to compact objects and cosmology. Either
outcome—refutation or confirmation—advances the field: DFD places gravitational physics on
a flat, optical foundation where every prediction is explicit, quantitative, and in reach of present
instruments.

Remaining limitations are not hidden but placed squarely on the table: the Boltzmann
hierarchy and full CMB/BAO perturbation theory are deferred but bounded by our µbg (a)
parametrization; cluster lensing is isolated to the kernel variant with Cavendish-scale tests;
the ψ ̸= ψ ′ coefficients are O(1) and directly calibratable by the LPI slope measurement; and
strong-field closure is already benchmarked to NICER and EHT tolerances with a well-posed
DFD–TOV scheme. Each of these is a finite, testable target. Thus the framework is not only
falsifiable in principle but operationally constrained today, with every open vulnerability tied to
an explicit experimental or numerical roadmap.

A

Light bending derivation

For spherically symmetric n(r), the conserved impact parameter is b = n(r) r sin θ. The ray
equation is
dθ
b
.
(39)
= √
2
dr
r n r 2 − b2
The total deflection is
Z ∞
b
√
α=2
dr − π,
(40)
2 2
2
r0 r n r − b


with r0 the distance of closest approach. For n(r) = exp 2GM/(rc2 ) , expansion yields
α≃

4GM
,
bc2

(41)

matching GR. Detailed derivations appear in [4, 7].

B

Matter-wave T 3 phase and parity



R ψ
2 /ℏ)
The phase
is
proportional
to
action,
∆ϕ
=
(mc
e
−
1
dt. Expanding ψ(z) = gz/c2 +


1
2 2
2 ∂g/∂z (z /c ) + . . . and integrating over fountain trajectories yields
∆ϕ = keff g T 2 +

keff ∂g 3
T + ...
2c2 ∂z

14

(42)

Parity (even in keff , rotation-odd). For an idealized vertical fountain with symmetric
up/down arms, denote the gradient-induced cubic contribution by βT 3 on the ascending leg
and −βT 3 on the descending leg when the rotation sense (or effective Coriolis projection) is
reversed:
∆ϕ↑ = +βT 3 + · · · ,
⇒

∆ϕ↓ = −βT 3 + · · · ,

∆ϕtotal = ∆ϕ↑ − ∆ϕ↓ = 2βT 3 + · · · .

Because the term arises from ∂g/∂z rather than the laser momentum transfer itself, it is even
under keff → −keff (while Coriolis reversals flip the sign). Numerically, near Earth ∂g/∂z ∼
3 × 10−6 s−2 gives ∆ϕT 3 ∼ 10−2 rad for T = 1 s, within reach of modern interferometers
[12, 13, 14, 15, 16].

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